Mathematical Colloquium The geometry of curves in an algebraic torus

A fundamental question in classical complex geometry concerns the types of meromorphic functions that a Riemann surface can admit. It is underlying question behind the Riemann-Roch theorem, and its study leads naturally to the discovery of the Jacobian of a curve. When these same questions are asked for families of Riemann surfaces, one is led quickly to a study of the space of punctured Riemann surfaces in an algebraic torus. I will discuss the geometry of this latter space. I will try to give a sense for how an understanding of its geometry leads to new results in areas of classical interest, such as Brill-Noether theory, as well as contemporary ones, such as Gromov-Witten theory. The story is held together by a rich underlying combinatorial structure, which is part of tropical geometry.